Finally, for \(\psi =\pm \frac{\pi }{2}\) , we find that the scattering diagram drastically simplifies. Indeed,
the geometric rays \(\Im [Z_\tau (\gamma )]=0\)  for \(r\ne 0\)  reduce to the contour lines
\(s=\frac{d}{r}\)  of the function \(s=\frac{\Im T_D}{\Im T}\)  defined in (REF ) (for \(r=0\) ,
the geometric rays \(\mathcal {R}^{\rm geo}_{\pi /2}(\gamma )\)  are empty). Hence,
scattering can only take place at the orbifold point \(\tau _o\)  and its images under \(\Gamma _1(3)\) , where the function \(s\)  is ill-defined. At each orbifold point, there are three incoming rays associated to the objects of corresponding exceptional collection, which scatter all at once according to the orbifold scattering diagram \(\mathcal {D}_o\) . In particular, a ray emitted from
the orbifold point \(\tau _o+m\)  into the fundamental domain \(\mathcal {F}_o(m)\) 
will escape to \(\tau =I\infty \)  in the range \(m-1<\tau _1<m\)  without encountering any wall of marginal stability.
For \(m=0\) , this
explains why the index for the orbifold quiver in the anti-attractor chamber \(\Omega _c(\gamma )\) 
agrees with the Gieseker index \(\Omega _\infty (\gamma )\)  for normalized torsion free
sheaves, as observed in [1]}, [2]}.
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